Decimals
Section: Number and Calculation | Syllabus: Cambridge Lower Secondary Checkpoint Mathematics (0862)
Understanding Decimals
A decimal contains a decimal point, splitting the number into a whole-number part on the left and a fractional part on the right. Each column - both sides of the point - is worth ten times the column to its right.
| Thousands | Hundreds | Tens | Ones | ● | Tenths | Hundredths | Thousandths |
|---|---|---|---|---|---|---|---|
| 1000 | 100 | 10 | 1 | ● | 1/10 = 0.1 | 1/100 = 0.01 | 1/1000 = 0.001 |
| 3 | 2 | 4 | 7 | ● | 5 | 6 | 8 |
3247.568 = 3000 + 200 + 40 + 7 + 0.5 + 0.06 + 0.008
Comparing and Ordering Decimals
- Line up the decimal points.
- Add zeros to give all the numbers the same number of decimal places, if needed - this doesn't change their value.
- Compare the digits column by column, from left to right.
Writing 0.6 as 0.60 makes it easy to see it is larger than 0.58
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Question: Which is larger: 0.6 or 0.58?
- Step 1: Write with the same number of decimal places: 0.60 and 0.58
- Step 2: Compare: 0.60 > 0.58
- Answer: 0.6 is larger
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Question: Arrange in ascending order: 3.45, 3.5, 3.405, 3.054.
- Step 1: Write with the same number of decimal places: 3.450, 3.500, 3.405, 3.054
- Step 2: Compare: 3.054 < 3.405 < 3.450 < 3.500
- Answer: 3.054, 3.405, 3.45, 3.5
Adding and Subtracting Decimals
Always line up the decimal points vertically before adding or subtracting, adding trailing zeros so both numbers have the same number of decimal places.
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Question: Calculate 23.45 + 7.8.
- Working: 23.45 + 7.80 (add a zero to help align) ------- 31.25
- Answer: 31.25
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Question: Calculate 15.7 − 8.45.
- Working: 15.70 (add a zero to help align) - 8.45 ------- 7.25
- Answer: 7.25
Multiplying Decimals
- Ignore the decimal points and multiply as if both numbers were whole numbers.
- Count the total number of decimal places in both numbers combined.
- Place the decimal point in the answer so it has that same total number of decimal places.
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Question: Calculate 3.2 × 1.5.
- Step 1: Multiply ignoring the decimals: 32 × 15 = 480
- Step 2: Count decimal places: 3.2 has 1, 1.5 has 1, so 2 total
- Step 3: Place the point 2 digits from the right: 4.80
- Answer: 4.8
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Question: Calculate 0.04 × 0.3.
- Step 1: Multiply ignoring the decimals: 4 × 3 = 12
- Step 2: Count decimal places: 0.04 has 2, 0.3 has 1, so 3 total
- Step 3: Place the point 3 digits from the right: 0.012
- Answer: 0.012
Multiplying by Powers of 10
| Operation | Rule | Example |
|---|---|---|
| × 10 | Move decimal point 1 place RIGHT | 4.56 × 10 = 45.6 |
| × 100 | Move decimal point 2 places RIGHT | 4.56 × 100 = 456 |
| × 1000 | Move decimal point 3 places RIGHT | 4.56 × 1000 = 4560 |
Multiplying and Dividing by 0.1, 0.01, 0.001
Since 0.1 = 10-1, 0.01 = 10-2, and 0.001 = 10-3, multiplying or dividing by these decimals is directly linked to multiplying or dividing by powers of 10 - just in the OPPOSITE direction to what you might expect:
- Key Point 1: Multiplying by 0.1, 0.01, 0.001, ... is the SAME as dividing by 10, 100, 1000, ...
- Key Point 2: Dividing by 0.1, 0.01, 0.001, ... is the SAME as multiplying by 10, 100, 1000, ...
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Question: Work out 60 × 0.1.
- Step 1: Multiplying by 0.1 is the same as dividing by 10
- Step 2: 60 ÷ 10
- Answer: 6
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Question: Work out 14 ÷ 0.01.
- Step 1: Dividing by 0.01 is the same as multiplying by 100
- Step 2: 14 × 100
- Answer: 1400
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Question: Work out 0.45 ÷ 0.001.
- Step 1: Dividing by 0.001 is the same as multiplying by 1000
- Step 2: 0.45 × 1000
- Answer: 450
Using a Known Calculation to Deduce Others
If you know one decimal calculation, you can deduce related ones instantly by tracking how each factor's place value has changed - the digits stay the same, only the decimal point shifts.
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Question: Given that 24 × 13 = 312, deduce the answers to 2.4 × 13, 24 × 1.3, 2.4 × 1.3, and 0.24 × 0.13.
- 2.4 × 13: 2.4 is 24 ÷ 10, so the answer is 312 ÷ 10 = 31.2
- 24 × 1.3: 1.3 is 13 ÷ 10, so the answer is also 312 ÷ 10 = 31.2
- 2.4 × 1.3: BOTH factors are ÷10, so the answer is 312 ÷ 10 ÷ 10 = 3.12
- 0.24 × 0.13: both factors are ÷100 from the original, so the answer is 312 ÷ 100 ÷ 100 = 0.0312
Dividing Decimals
Dividing by Whole Numbers
Divide as normal, keeping the decimal point in the same column position.
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Question: Calculate 8.4 ÷ 2.
- Step 1: Divide as normal, keeping the point in place: 8.4 ÷ 2 = 4.2
- Answer: 4.2
Dividing by Decimals
- Make the divisor (the number you're dividing by) a whole number by multiplying both numbers by 10, 100, or 1000.
- Then divide as normal.
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Question: Calculate 6.3 ÷ 0.7.
- Step 1: Multiply both numbers by 10 to clear the divisor's decimal: 63 ÷ 7
- Step 2: 63 ÷ 7 = 9
- Answer: 9
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Question: Calculate 1.2 ÷ 0.04.
- Step 1: Multiply both numbers by 100 to clear the divisor's decimal: 120 ÷ 4
- Step 2: 120 ÷ 4 = 30
- Answer: 30
Dividing by Powers of 10
| Operation | Rule | Example |
|---|---|---|
| ÷ 10 | Move decimal point 1 place LEFT | 45.6 ÷ 10 = 4.56 |
| ÷ 100 | Move decimal point 2 places LEFT | 456 ÷ 100 = 4.56 |
| ÷ 1000 | Move decimal point 3 places LEFT | 4560 ÷ 1000 = 4.56 |
Converting Fractions to Decimals
Divide the numerator by the denominator to convert any fraction to a decimal.
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